Geodesic
On balanced bases
Matematičeskie zametki, Tome 77 (2005) no. 2, pp. 213-218.

Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that either a given balanced basis of the algebra $(n+1)M_1\oplus M_n$ or the corresponding complementary basis is of rank $n+1$. This result enables us to claim that the algebra $(n+1)M_1\oplus M_n$ is balanced if and only if the matrix algebra $M_n$ admits a WP-decomposition, i.e., a family of $n+1$ subalgebras conjugate to the diagonal algebra and such that any two algebras in this family intersect orthogonally (with respect to the form $\operatorname{tr}XY$) and their intersection is the trivial subalgebra. Thus, the problem of whether or not the algebra $(n+1)M_1\oplus M_n$ is balanced is equivalent to the well-known Winnie-the-Pooh problem on the existence of an orthogonal decomposition of a simple Lie algebra of type $A_{n-1}$ into the sum of Cartan subalgebras. 
@article{MZM_2005_77_2_a4,
     author = {D. N. Ivanov},
     title = {On balanced bases},
     journal = {Matemati\v{c}eskie zametki},
     pages = {213--218},
     publisher = {mathdoc},
     volume = {77},
     number = {2},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2005_77_2_a4/}
}
TY  - JOUR
AU  - D. N. Ivanov
TI  - On balanced bases
JO  - Matematičeskie zametki
PY  - 2005
SP  - 213
EP  - 218
VL  - 77
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2005_77_2_a4/
LA  - ru
ID  - MZM_2005_77_2_a4
ER  - 
%0 Journal Article
%A D. N. Ivanov
%T On balanced bases
%J Matematičeskie zametki
%D 2005
%P 213-218
%V 77
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2005_77_2_a4/
%G ru
%F MZM_2005_77_2_a4
D. N. Ivanov. On balanced bases. Matematičeskie zametki, Tome 77 (2005) no. 2, pp. 213-218. http://geodesic.mathdoc.fr/item/MZM_2005_77_2_a4/

[1] Ivanov D. N., “Ortogonalnye razlozheniya assotsiativnykh algebr i sbalansirovannye sistemy idempotentov”, Matem. sb., 189:12 (1998), 83–102 | MR | Zbl

[2] Kostrikin A. I., Kostrikin I. A., Ufnarovskii V. A., “Ortogonalnye razlozheniya prostykh algebr Li tipa $A_n$”, Tr. MIAN, 158, Nauka, M., 1981, 105–120 | MR | Zbl

[3] Beth T., Jungnickel D., Lenz H., Design Theory, 2nd edition, Cambridge University Press, Cambridge, 1999 | MR

[4] Ivanov D. N., “Ortogonalnye razlozheniya poluprostykh assotsiativnykh algebr”, Vestn. MGU. Ser. 1. Matem., mekh., 43:1 (1988), 9–14 | MR | Zbl

[5] Kostrikin A. I., Pham Huu Tiep, Orthogonal Decompositions and Integral Lattices, Walter de Gruyter, Berlin, 1994 | MR

[6] Ivanov D. N., “Analog teoremy Vagnera dlya ortogonalnykh razlozhenii algebry matrits $M_n(\mathbb C)$”, UMN, 49:1 (1994), 215–216 | MR | Zbl

[7] Bahturin Yu. A., Sehgal S. K., Zaicev M. V., “Groups gradings on associative algebras”, J. Algebra, 241 (2001), 677–698 | DOI | MR | Zbl

[8] Ivanov D. N., “Sbalansirovannye sistemy iz primitivnykh idempotentov v algebrakh matrits”, Matem. sb., 191:4 (2000), 67–90 | Zbl

[9] Ivanov D. N., “O sbalansirovannykh sistemakh idempotentov”, Matem. sb., 192:4 (2001), 73–86 | Zbl

[10] Ivanov D. N., “Orthogonal decompositions and idempotent configurations in semisimple associative algebras”, Comm. Algebra, 29:9 (2001), 3839–3887 | DOI | Zbl